A combination theorem for affine tree-free groups

Let $\Lambda_0$ be an ordered abelian group. We show how an $\mathrm{ATF}(\mathbb{Z}\times\Lambda_0)$ group -- that is, a group admitting a free affine action without inversions on a $\mathbb{Z}\times\Lambda_0$-tree -- admits a natural graph of groups decomposition, where vertex groups inherit actions on $\Lambda_0$-trees. Using recent work of various authors, it follows that a finitely generated group admitting a free affine action on a $\mathbb{Z}^n$-tree where no line has its orientation reversed is relatively hyperbolic with nilpotent parabolics, is locally quasiconvex, and has solvable word, conjugacy and isomorphism problems. Conversely, given a graph of groups satisfying certain conditions, we show how an affine action of its fundamental group can be constructed. Specialising to the case of free affine actions, we obtain a large class of $\mathrm{ATF}(\mathbb{Z}\times\Lambda_0)$ groups that do not act freely by isometries on any $\Lambda_1$-tree. We also give an example of a group that admits a free isometric action on a $\mathbb{Z}\times\mathbb{Z}$-tree but which is not residually nilpotent.